In geometry, two shapes are similar when one is a dilation of the other.

Consider Dr. Evil and Mini Me from Mike Meyers’ hit movie Austin Powers. Mini Me is supposed to be an exact replica of Dr. Evil.

4 S IMILAR T RIANGLE N OTATION The symbol for similarity is The order of the letters is important: corresponding letters should name congruent angles. A B C E D F

 ABC   DEF When we say that triangles are similar there are several conclusions that come from it. A  DA  D B  EB  E C  FC  F AB DE BC EF AC DF = = 1. All angles are congruent 2. All sides are proportional.

2. SSS Similarity Theorem  3 pairs of proportional sides If we need to PROVE that a pair of triangles are similar how many of those statements do we need? We do not need all of them. There are three special combinations that we can use to prove similarity of triangles. 3. SAS Similarity Theorem  2 pairs of proportional sides and congruent angles between them 1. AA Similarity Theorem  2 pairs of congruent angles

1. AA Similarity Theorem  2 pairs of congruent angles M N O Q PR 70  50  m  N = m  R m  O = m  P  MNO   QRP

You may need to find the 3 rd angles in order to see if 2 pairs are congruent! 87  34  S T U X Y Z m  T = m  X m  S = 180  - (34  + 87  ) m  S = 180   m  S = 59  m  S = m  Z  TSU   XZY 59  34 

2. SSS Similarity Theorem  3 pairs of proportional sides A BC E F D  ABC   DFE The scale factor is 1.25

3. SAS Similarity Theorem  2 pairs of proportional sides and congruent angles between them G H I L JK  m  H = m  K  GHI   LKJ

SAS does not work unless angle is in between proportional sides! G H I L JK  Angles I and J do not fall in between sides GH and HI and sides LK and KJ respectively.

1. Mark the Given. 2. Mark … Shared Angles or Vertical Angles 3. Choose a Method. (AA, SSS, SAS) Think about what you need for the chosen method and be sure to include those parts in the proof. Steps for proving triangles similar: Proving Triangles Similar

CAN YOU PROVE THESE ARE SIMILAR? WHY OR WHY NOT???

Mary is 5 ft 6 inches tall. She casts a 2 foot shadow. The tree casts a 7 foot shadow. How tall is the tree?

Mary is 5 ft 6 inches tall. She casts a 2 foot shadow. The tree casts a 7 foot shadow. How tall is the tree? Mary’s height Tree’s height Mary’s shadow Tree’s shadow x =

Mary is 5 ft 6 inches tall. She casts a 2 foot shadow. The tree casts a 7 foot shadow. How tall is the tree? Mary’s height Tree’s height Mary’s shadow Tree’s shadow x = x 2 7 =

x 2 7 = 7 ( 5.5 ) = 2 x 38.5 = 2 x x = The height of the tree is feet